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There are many possible generalizations of the theorem of Pythagoras. Some of the most well known ones are the cosine formula, the distance in n dimensions, and Ptolemy’s theorem. These three generalisations, along with seven others, are discussed in De Villiers (2009, pp. 69-75).
Pythagoras's Proof. Given any right triangle with legs \( a \) and \(b \) and hypotenuse \( c\) like the above, use four of them to make a square with sides \( a+b\) as shown below: This forms a square in the center with side length \( c \) and thus an area of \( c^2.
Thabit's Generalization of the Theorem of Pythagoras. Explanation (proof): Can you explain why (prove) Thabit's generalization above is true? Further Exploration: What are the respective conditions under which BC is greater than or smaller than BD + EC?
Proof of Thabit ibn Qurra’s Theorem. It will be helpful to translate everything into algebra by introducing the following notation: |∠∠∠B′′′′AB| = δδδδ111 |∠∠∠∠C′′′′AX| = δδδδ2222 We then have the following equations, the first of which is true by the additivity properties of
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12 Δεκ 2019 · The Corbettmaths Practice Questions on Geometric Proof for Level 2 Further Maths.
If you think of a 2 +b 2 =c 2 as the geometrical result that the sum of areas of squares constructed with sides a and b is the area of a square placed on c, then the Pythagorean theorem is true not just for constructing squares on the sides, but any similar figures.
